Leaving Certificate Examination 1965 Honours Applied Mathematics
Question 1
Two pegs are fixed at points 
and 
in the same horizontal line. One end of a light string is attached to and the other end to . When masses of 
and 
lb. are attached to points 
and 
, respectively, on the string, the 
and 
. Find the tension, in lb. wt., in each of the segments 
, 
, 
of the string, correct in each case to one significant figure.
Question 2
Three particles weighing 
, 
, 
are placed, respectively, at points the coordinates of which are 
, 
, 
. If 
are the coordinates of the centre of gravity of the system, show that
, 
Particles of weight 
, 
, 
and 
grams are placed, respectively, at the points , , , , the coordinate of which are 
, 
, 
, 
respectively. Find the position of the centre of gravity of the system.
What additional weight must be placed at if the centre of gravity is required to be at the mid-point of 
?
Question 3
What is meant by the velocity of a particle relative to another particle?
Explain, with the aid of a diagram, how the velocity of relative to may be found if the velocities of and are known.
The velocity of a ship relative to a steady wind is 
m.p.h. in the direction 
North of East and the velocity of a boat relative to the same wind is 
m.p.h. in the direction 
South of West. Find the velocity (in magnitude and direction) of the ship relative to the boat.
Question 4
A car weighing 
cwt. is descending an incline of in 
. The speed of the car is 
m.p.h. and it is accelerating at the rate of 
ft. per sec.
. If the frictional resistance to motion is equivalent to lb. wt., find the horse-power at which the car is working.
If the car travels in a straight line on a horizontal track against a frictional resistance to motion of 
lb. wt., find the greatest speed it attains if it develops horse-power.
Question 5
If three forces acting at a point are in equilibrium, prove that each force is proportional to the sine of the angles between the lines of action of the other two forces. The perpendiculars drawn from the vertices of an acute – angled triangle meet at 
. The perpendiculars from to the sides of the triangles are the lines of action of three forces. Each force acts away from and is proportional to the length of the side to which it is perpendicular. Prove that the three forces are in equilibrium.
Question 6
Derive an expression, in terms of the angle of projection and the initial velocity, for (i) the range, (ii) the greatest height reached by a projectile.
Two particles are projected from a point with the same initial velocity at angles of elevation 
and 
. If 
, show that the range of each of the two particles is the same.
If the initial velocity of each of the particles if 
ft. per second and 
ft. is the greatest height reached by one of the particles, find the greatest height reached by the other particle.
Question 7
Define simple harmonic motion.
A particle is moving in a striaght line with simple harmonic motion. When it is cm. from its mean position, its velocity and acceleration are cm. per sec. and cm. per sec., respectively. Find
(i) the greatest velocity of the particle,
(ii) the period of the motion,
(iii) the average velocity of the particle as it travels from rest to rest.
Question 8
A circular piece of tin rotates at the rate of revolutions per minute about an axis through its centre , the axis being perpendicular to the plane of the tin. and are two poitns on the tin such that 
inches, 
ft. 
ins. and 
. Weights of lb. and lb. are placed, respectively, at and . Find the magnitude and direction of the resultant centrifugal force on the axis.
If is a point on the tin such that 
ft., 
and 
, what weight must be placed at if the resultant centrifugal force is to act along 
?
Question 9
A vessel in the shape of a cube has internal edges each ft. long. A liquid fills the vessel to a height of inches and a second liquid which does not mix the the first occupies the remainder of the vessel. On each vertical side the total thrust on that part in contact with one liquid is equal to the total thrust on that part in contact with the other. Show that the specific gravities of the two liquids are in the ratio 
.
Citation:
State Examinations Commission (2018). State Examination Commission. Accessed at: https://www.examinations.ie/?l=en&mc=au&sc=ru
Malone, D and Murray, H. (2016). Archive of Maths State Exams Papers. Accessed at: http://archive.maths.nuim.ie/staff/dmalone/StateExamPapers/
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